CS5801 Coursework Template Proforma

# Add code here to load all the required libraries with `library()`.  
# Do not include any `install.package()` for any required packages in this rmd file.

library(ggplot2)
library(readr)
library(VIM)
library(mice)
library(vcdExtra)
library(car)
library(tidyverse)

1. Organise and clean the data

1.1 Subset the data into the specific dataset allocated

# Only change the value for SID 
# Assign your student id into the variable SID, for example:
SID <- 2330859                  # This is an example, replace 2101234 with your actual ID
SIDoffset <- (SID %% 50) + 1    # Your SID mod 50 + 1

load("car-analysis-data.Rda")
# Now subset the car data set
# Pick every 50th observation starting from your offset
# Put into your data frame named mydf (you can rename it)
mydf <- cars.analysis[seq(from=SIDoffset,to=nrow(cars.analysis),by=50),]

1.2 Data quality analysis plan

I will find any NA or implausible values. These are significant because analyses could be affected by missing data – i.e. a trend line on a graph may not represent the true nature of the data. Implausible values can create outliers which also affects the regression line in modelling, still, outliers are plausible therefore implausibility must be well justified. Appropriate imputation methods will be used for variables with NA values starting with the variable with the highest number of NA values. I will also check that any implausible value was not chosen by the data owners for arbitrary imputation by checking the frequency of the value compared to other values. In this case, I would explore further to understand what the value meant and better understand how to impute the value. To keep as much of the data as possible, I will impute unknown values where appropriate. I will also check the levels in the categorical variables to check for misspellings and correct erroneous levels’ names. I will then convert all variables apart from price and mileage into factors because unlike price and mileage they are either binary, categorical or fixed values such as brand.

1.3 Data quality analysis findings

# Create a new data frame to store the transformations
mydf.comp <- mydf

# Check for any NAs
no.NA <- sum(is.na(mydf.comp))                  # Gives a sum of the total NAs in the data frame
paste("There are", no.NA,"NA values in the data. They can be found in the following variables:")

[1] "There are 124 NA values in the data. They can be found in the following variables:"

colSums(is.na(mydf.comp))                       # Shows how many NAs are in each variable

                 brand                   year                mileage            engine_size automatic_transmission 
                     0                      0                      0                     19                      0 
                  fuel             drivetrain                min_mpg                max_mpg                damaged 
                     0                      0                     49                     49                      2 
           first_owner      navigation_system              bluetooth      third_row_seating           heated_seats 
                     5                      0                      0                      0                      0 
                 price 
                     0 

aggr(mydf)                                      # Visualizes the missing data in the data

Most NAs are in the mpg variables.

# Impute max_mpg's NA values using its median 

median_max_mpg <- median(mydf.comp$max_mpg[mydf.comp$max_mpg > 0], na.rm = TRUE)  # Find the median (excluding NA and negative values)

mydf.comp$max_mpg[is.na(mydf.comp$max_mpg)] = median_max_mpg                      # Replace NA values in max_mpg with the median

# Compare its pre-imputation distribution to its post
ggplot(data=mydf, aes(x=max_mpg)) + geom_histogram(bins = 10) + theme_linedraw() +ggtitle("Histogram of max_mpg without imputed median for NA values") + xlab("Max_mpg without imputed values")

ggplot(data=mydf.comp, aes(x=max_mpg)) + geom_histogram(bins = 10) + theme_linedraw() +ggtitle("Histogram of max_mpg with imputed median for NA values") + xlab("Max_mpg with imputed values") 

This affects the distribution significantly. I will try Multiple Imputation by Chained Equations.

# Impute all NAs in the data using mice
mydf.mice <- mice(mydf)

 iter imp variable
  1   1  engine_size  min_mpg  max_mpg  damaged  first_owner
  1   2  engine_size  min_mpg  max_mpg  damaged  first_owner
  1   3  engine_size  min_mpg  max_mpg  damaged  first_owner
  1   4  engine_size  min_mpg  max_mpg  damaged  first_owner
  1   5  engine_size  min_mpg  max_mpg  damaged  first_owner
  2   1  engine_size  min_mpg  max_mpg  damaged  first_owner
  2   2  engine_size  min_mpg  max_mpg  damaged  first_owner
  2   3  engine_size  min_mpg  max_mpg  damaged  first_owner
  2   4  engine_size  min_mpg  max_mpg  damaged  first_owner
  2   5  engine_size  min_mpg  max_mpg  damaged  first_owner
  3   1  engine_size  min_mpg  max_mpg  damaged  first_owner
  3   2  engine_size  min_mpg  max_mpg  damaged  first_owner
  3   3  engine_size  min_mpg  max_mpg  damaged  first_owner
  3   4  engine_size  min_mpg  max_mpg  damaged  first_owner
  3   5  engine_size  min_mpg  max_mpg  damaged  first_owner
  4   1  engine_size  min_mpg  max_mpg  damaged  first_owner
  4   2  engine_size  min_mpg  max_mpg  damaged  first_owner
  4   3  engine_size  min_mpg  max_mpg  damaged  first_owner
  4   4  engine_size  min_mpg  max_mpg  damaged  first_owner
  4   5  engine_size  min_mpg  max_mpg  damaged  first_owner
  5   1  engine_size  min_mpg  max_mpg  damaged  first_owner
  5   2  engine_size  min_mpg  max_mpg  damaged  first_owner
  5   3  engine_size  min_mpg  max_mpg  damaged  first_owner
  5   4  engine_size  min_mpg  max_mpg  damaged  first_owner
  5   5  engine_size  min_mpg  max_mpg  damaged  first_owner

Warning: Number of logged events: 3

mydf.comp <- complete(mydf.mice)

ggplot(data=mydf, aes(x=max_mpg)) + geom_histogram(bins = 10) + theme_linedraw() +ggtitle("Histogram of max_mpg without imputed 'mice' for NA values") + xlab("Max_mpg without imputed values")

ggplot(data=mydf.comp, aes(x=max_mpg)) + geom_histogram(bins = 10) + theme_linedraw() +ggtitle("Histogram of max_mpg with imputed 'mice' for NA values") + xlab("Max_mpg with imputed values")

This method fits better.

# Statistically summarize the data
summary(mydf.comp)

    brand                year         mileage        engine_size      automatic_transmission     fuel          
 Length:410         Min.   :1964   Min.   :     0   Min.   :  1.200   Min.   :0.0000         Length:410        
 Class :character   1st Qu.:2015   1st Qu.: 22286   1st Qu.:  2.000   1st Qu.:1.0000         Class :character  
 Mode  :character   Median :2018   Median : 44394   Median :  2.450   Median :1.0000         Mode  :character  
                    Mean   :2017   Mean   : 51588   Mean   :  3.689   Mean   :0.9146                           
                    3rd Qu.:2021   3rd Qu.: 72747   3rd Qu.:  3.500   3rd Qu.:1.0000                           
                    Max.   :2023   Max.   :209968   Max.   :390.000   Max.   :1.0000                           
  drivetrain           min_mpg         max_mpg          damaged        first_owner     navigation_system
 Length:410         Min.   : 0.00   Min.   :-30.00   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   
 Class :character   1st Qu.:18.00   1st Qu.: 24.00   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   
 Mode  :character   Median :21.50   Median : 28.00   Median :0.0000   Median :0.0000   Median :0.0000   
                    Mean   :21.53   Mean   : 28.25   Mean   :0.2659   Mean   :0.4902   Mean   :0.4707   
                    3rd Qu.:24.00   3rd Qu.: 32.00   3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000   
                    Max.   :47.00   Max.   : 51.00   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   
   bluetooth      third_row_seating  heated_seats        price      
 Min.   :0.0000   Min.   :0.0000    Min.   :0.0000   Min.   : 4500  
 1st Qu.:1.0000   1st Qu.:0.0000    1st Qu.:0.0000   1st Qu.:18988  
 Median :1.0000   Median :0.0000    Median :1.0000   Median :29370  
 Mean   :0.8854   Mean   :0.1537    Mean   :0.5098   Mean   :28944  
 3rd Qu.:1.0000   3rd Qu.:0.0000    3rd Qu.:1.0000   3rd Qu.:37998  
 Max.   :1.0000   Max.   :1.0000    Max.   :1.0000   Max.   :54999  

table(mydf.comp[mydf.comp < 0])

-30 
  2 

Year, mileage and price values all seem plausible. Negative values in max_mpg are implausible. It may be representative. I will verify using its frequency in cars.analysis.

# Check if and how frequently -30 were used in cars.analysis
summary(cars.analysis)

    brand                year         mileage        engine_size      automatic_transmission     fuel          
 Length:20483       Min.   :1953   Min.   :     0   Min.   :  0.000   Min.   :0.0000         Length:20483      
 Class :character   1st Qu.:2016   1st Qu.: 23582   1st Qu.:  2.000   1st Qu.:1.0000         Class :character  
 Mode  :character   Median :2019   Median : 44709   Median :  2.400   Median :1.0000         Mode  :character  
                    Mean   :2017   Mean   : 52575   Mean   :  2.821   Mean   :0.9149                           
                    3rd Qu.:2020   3rd Qu.: 73894   3rd Qu.:  3.500   3rd Qu.:1.0000                           
                    Max.   :2023   Max.   :383614   Max.   :390.000   Max.   :1.0000                           
                                                    NA's   :1114                                               
  drivetrain           min_mpg         max_mpg        damaged        first_owner     navigation_system
 Length:20483       Min.   : 0.00   Min.   :-30    Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   
 Class :character   1st Qu.:18.00   1st Qu.: 24    1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   
 Mode  :character   Median :21.00   Median : 28    Median :0.0000   Median :0.0000   Median :0.0000   
                    Mean   :21.29   Mean   : 28    Mean   :0.2412   Mean   :0.4952   Mean   :0.4485   
                    3rd Qu.:24.00   3rd Qu.: 32    3rd Qu.:0.0000   3rd Qu.:1.0000   3rd Qu.:1.0000   
                    Max.   :89.00   Max.   :100    Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   
                    NA's   :2987    NA's   :2970   NA's   :204      NA's   :308                       
   bluetooth      third_row_seating  heated_seats        price      
 Min.   :0.0000   Min.   :0.0000    Min.   :0.0000   Min.   : 1495  
 1st Qu.:1.0000   1st Qu.:0.0000    1st Qu.:0.0000   1st Qu.:18995  
 Median :1.0000   Median :0.0000    Median :0.0000   Median :27998  
 Mean   :0.8789   Mean   :0.1445    Mean   :0.4678   Mean   :28458  
 3rd Qu.:1.0000   3rd Qu.:0.0000    3rd Qu.:1.0000   3rd Qu.:37000  
 Max.   :1.0000   Max.   :1.0000    Max.   :1.0000   Max.   :54999  
                                                                    

table(cars.analysis[cars.analysis < 0])

-30.0 
  101 

101 ‘-30’ observations suggest it’s a code.

# Check for other arbitrary imputations
table(cars.analysis$max_mpg)

 -30    0   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27 27.5   28   29   30 
 101   91    1    3    7   14   22   30  151  267  388  401  667  650  927 1125 1002 1052 1075    1 1452 1028 1333 
  31   32   33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48   49   50   51   52   53 
1146  662  746  676  591  328  337  348  143  374  100   39   48   27    5    6   10   19    8    1   34    4   11 
  54   55   56   57   58   59   60   61   71   80   94  100 
   5   19    2    1    2    8    3    1    1   16    1    3 

table(mydf.comp$max_mpg)

-30   0  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  43 
  2   4   1   5   5  12   7  16  11  25  20  10  33  25  45  14  30  25  20  12  20  18   9   6   7   2  15   4   3 
 47  48  51 
  1   1   2 

0 mpg means the car is stationary. I will combine -30 to 0 as code for: cars that do not run.

# Convert negative observations in the max_mpg variable to 0
mydf.comp$max_mpg[mydf.comp$max_mpg < 0] = 0
table(mydf.comp$max_mpg)                          # Check that there are now eight 0 max_mpg values

 0 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 43 47 48 51 
 6  1  5  5 12  7 16 11 25 20 10 33 25 45 14 30 25 20 12 20 18  9  6  7  2 15  4  3  1  1  2 

A 390-litre car engine size is implausible.

# Find out if there are other values as large
table(mydf.comp$engine_size)

1.2 1.3 1.4 1.5 1.6 1.8   2 2.3 2.4 2.5 2.9   3 3.3 3.4 3.5 3.6 3.7 3.8   4 4.2 4.4 4.6 4.7 4.8   5 5.3 5.5 5.6 5.7 
  5   1  22  13  14  10 120   2  18  36   1  43   8   1  33  23   3   5   7   1   2   6   6   1  10   2   1   5   2 
  6 6.2 6.8 390 
  3   4   1   1 

# Change 390 to 3.9
mydf.comp$engine_size[mydf.comp$engine_size==390] = 3.9

summary(mydf.comp$engine_size)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.200   2.000   2.450   2.748   3.500   6.800 

# Identify the levels of 'fuel'
table(mydf.comp$fuel)                # 'Pertol' is a misspelling

  Diesel Electric      GPL   Hybrid   Pertol   Petrol  Unknown 
       5        8        3       26        2      364        2 

# Change the misspelled 'pertol' to the 'petrol' level
mydf.comp[mydf.comp$fuel == "Pertol", "fuel" ] = "Petrol"

# Impute 'Unknown' values using Frequent Category Imputation
mydf.comp$fuel[(mydf.comp$fuel=="Unknown")] <- names(which.max(table(mydf.comp$fuel)))

table(mydf.comp$fuel)

  Diesel Electric      GPL   Hybrid   Petrol 
       5        8        3       26      368 

# Check other categorical variable levels for errors
table(mydf.comp$brand)

         Alfa          Audi           BMW      Cadillac     Chevrolet          FIAT          Ford         Honda 
           24            21            17            19            12            24            13            24 
      Hyundai        Jaguar          Jeep           Kia          Land         Lexus      Maserati         Mazda 
           18            14            11            20             9            15            12            21 
Mercedes-Benz          MINI    Mitsubishi        Nissan       Porsche        Suzuki        Toyota    Volkswagen 
           15            21            24            13             9             2            10            21 
        Volvo 
           21 

table(mydf.comp$drivetrain) 

 Four-wheel Drive Front-wheel Drive  Rear-wheel Drive           Unknown 
              225               119                64                 2 

# Impute 'Unknown' values using Frequent Category Imputation
mydf.comp$drivetrain[(mydf.comp$drivetrain=="Unknown")] <- names(which.max(table(mydf.comp$drivetrain)))

table(mydf.comp$drivetrain)

 Four-wheel Drive Front-wheel Drive  Rear-wheel Drive 
              227               119                64 

Categorical and binary should be factors and numerical should be numerical for EDA.

# Check to see if variables have been read in correctly
str(mydf.comp)

'data.frame':   410 obs. of  16 variables:
 $ brand                 : chr  "Maserati" "Jeep" "Toyota" "Land" ...
 $ year                  : num  2018 2021 2020 2015 2021 ...
 $ mileage               : num  19267 36433 35134 106110 48435 ...
 $ engine_size           : num  3 3.6 2.5 3 2.5 1.5 3 2 2 3.3 ...
 $ automatic_transmission: num  1 1 1 1 1 1 1 1 1 1 ...
 $ fuel                  : chr  "Petrol" "Petrol" "Hybrid" "Petrol" ...
 $ drivetrain            : chr  "Rear-wheel Drive" "Four-wheel Drive" "Four-wheel Drive" "Four-wheel Drive" ...
 $ min_mpg               : num  17 21 34 14 35 27 18 17 22 22 ...
 $ max_mpg               : num  24 28 35 19 36 36 23 23 28 29 ...
 $ damaged               : num  1 0 0 0 0 0 0 0 0 0 ...
 $ first_owner           : num  1 1 1 0 1 1 1 1 0 0 ...
 $ navigation_system     : num  1 0 1 0 1 0 1 0 1 0 ...
 $ bluetooth             : num  1 1 1 1 1 1 1 1 1 1 ...
 $ third_row_seating     : num  0 0 1 0 1 0 0 0 0 1 ...
 $ heated_seats          : num  1 0 1 0 1 1 0 0 1 0 ...
 $ price                 : num  37699 47288 45698 19888 50995 ...

# Convert all variables into appropriate data types

columns <- names(mydf.comp)                                            
mydf.comp[, columns] <- lapply(mydf.comp[,columns] , factor)    # Converts all columns into a factor

mydf.comp$price <-as.numeric(mydf$price)                        # Convert numeric vectors from factor to numeric
mydf.comp$mileage <- as.numeric(mydf$mileage)

mydf.comp$engine_size <- as.character(mydf.comp$engine_size)
mydf.comp$engine_size <- as.numeric(mydf.comp$engine_size)

mydf.comp$max_mpg <- as.character(mydf.comp$max_mpg)
mydf.comp$max_mpg <- as.numeric(mydf.comp$max_mpg)

mydf.comp$min_mpg <- as.character(mydf.comp$min_mpg)
mydf.comp$min_mpg <- as.numeric(mydf.comp$min_mpg)

mydf.comp$year <- as.character(mydf.comp$year)
mydf.comp$year <- as.numeric(mydf.comp$year)

str(mydf.comp)                                                 # Shows that each variable has been correctly read in

'data.frame':   410 obs. of  16 variables:
 $ brand                 : Factor w/ 25 levels "Alfa","Audi",..: 15 11 23 13 23 18 13 24 25 9 ...
 $ year                  : num  2018 2021 2020 2015 2021 ...
 $ mileage               : num  19267 36433 35134 106110 48435 ...
 $ engine_size           : num  3 3.6 2.5 3 2.5 1.5 3 2 2 3.3 ...
 $ automatic_transmission: Factor w/ 2 levels "0","1": 2 2 2 2 2 2 2 2 2 2 ...
 $ fuel                  : Factor w/ 5 levels "Diesel","Electric",..: 5 5 4 5 4 5 5 1 4 5 ...
 $ drivetrain            : Factor w/ 3 levels "Four-wheel Drive",..: 3 1 1 1 1 2 1 2 1 1 ...
 $ min_mpg               : num  17 21 34 14 35 27 18 17 22 22 ...
 $ max_mpg               : num  24 28 35 19 36 36 23 23 28 29 ...
 $ damaged               : Factor w/ 2 levels "0","1": 2 1 1 1 1 1 1 1 1 1 ...
 $ first_owner           : Factor w/ 2 levels "0","1": 2 2 2 1 2 2 2 2 1 1 ...
 $ navigation_system     : Factor w/ 2 levels "0","1": 2 1 2 1 2 1 2 1 2 1 ...
 $ bluetooth             : Factor w/ 2 levels "0","1": 2 2 2 2 2 2 2 2 2 2 ...
 $ third_row_seating     : Factor w/ 2 levels "0","1": 1 1 2 1 2 1 1 1 1 2 ...
 $ heated_seats          : Factor w/ 2 levels "0","1": 2 1 2 1 2 2 1 1 2 1 ...
 $ price                 : num  37699 47288 45698 19888 50995 ...

1.4 Data cleaning

There were 124 NAs which can bias analysis outcomes. After visualizing the effect of median imputation and Multiple Imputation by Chained Equations, the latter was chosen because it better matched the original distribution. There were no errors found in year, price or mileage because all seemed plausible and there was no missingness.

There was an implausible value (-30) in the max_mpg variable which I cross-referenced with the cars.analysis data to find 101 negative values, all -30. This suggests it’s an arbitrary imputation code perhaps suggesting the car is being sold for parts. Another code (0) was found and combined with -30 to represent malfunctioning cars. The reason is that 0 mpg means the car is stationary therefore cannot work, and thus is sold for parts.

A 390-litre engine size is implausible and was likely supposed to be 3.9, which is more in line with the variable mean of 3.7. I amended two misspellings in ‘fuel’ and imputed unknown observations using mode imputation in ‘fuel’ and ‘drivetrain’.

Finally, R read the binary variables incorrectly, so I converted all variables into factors except for the numerical variables.

2. Exploratory Data Analysis (EDA)

2.1 EDA plan

I will start by visualizing the relationship between mileage (numerical) and price (numerical) using a scatterplot because intuitively I know that generally, cars with more miles are cheaper to buy. If the graph suggests correlation I will justify it statistically using Fisher’s p test and the correlation coefficient, before running a linear regression model with both variables and checking that the Residuals and Q-Q plot look random and normal respectively. I will do the same for ‘year’ for the same reason. I will then visualize the binary variable relationships to price using boxplots. Finally, I will visualize the mpg variables using scatter plots as they are both numerical variables. To avoid heteroskedasticity later, I will plot histograms to compare and visualize multicollinearity. If there is any, I will transform the data using the natural logarithm.

2.2 EDA execution

# Visualize the relationship between mileage and price
ggplot(data=mydf.comp, aes(x=mileage, y=price)) + geom_point() + theme_linedraw() + ggtitle("Scatter plot of the relationship between price and mileage") + xlab("Price") + ylab("Mileage")

This scatter plot shows a negative relationship between price and mileage - the more mileage it has, the cheaper it is.

# Numerically justify the graph
cor.test(mydf.comp$mileage,mydf.comp$price)

    Pearson's product-moment correlation

data:  mydf.comp$mileage and mydf.comp$price
t = -16.225, df = 408, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.6817493 -0.5635850
sample estimates:
       cor 
-0.6262505 

The correlation value is closer to -1 which means the higher the X-axis value (Mileage), the lower the Y-axis value (Price). I can use a hypothesis test to confirm my finding: H_0:Price & Mileage are not correlated & H_1:Price & Mileage are corelated where \(\alpha=0.05\).

The p-value is: 2.2e-16 which is a lot smaller than alpha which means I have enough evidence to reject the null hypothesis. Price and mileage are correlated.

# Create a linear regression model of price and mileage
lm.price.mileage <- lm(formula = mydf.comp$price~mydf.comp$mileage)
summary(lm.price.mileage)

Call:
lm(formula = mydf.comp$price ~ mydf.comp$mileage)

Residuals:
     Min       1Q   Median       3Q      Max 
-27062.1  -7243.4   -560.3   6438.2  27026.3 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)        3.902e+04  7.733e+02   50.46   <2e-16 ***
mydf.comp$mileage -1.953e-01  1.204e-02  -16.23   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 9332 on 408 degrees of freedom
Multiple R-squared:  0.3922,    Adjusted R-squared:  0.3907 
F-statistic: 263.3 on 1 and 408 DF,  p-value: < 2.2e-16

plot(lm.price.mileage)

The R^2 however is smaller than 0.5 suggesting it is not significant. Still the p-value suggests they are dependent and the intercept suggests that for every unit change in mileage, price decreases by 2.008e+00. Visually, the Residual Vs Fitted looks like a funnel suggesting hetroskedasticity.

# Create a new variable for the log of 'price'
mydf.comp$price.log <- log(mydf.comp$price)

# Run the model again
lm.price.log <- lm(formula = mydf.comp$price.log~mydf.comp$mileage)
summary(lm.price.log)

Call:
lm(formula = mydf.comp$price.log ~ mydf.comp$mileage)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.22535 -0.23502  0.02798  0.26199  1.06008 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)        1.061e+01  3.027e-02  350.62   <2e-16 ***
mydf.comp$mileage -8.657e-06  4.712e-07  -18.37   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3653 on 408 degrees of freedom
Multiple R-squared:  0.4527,    Adjusted R-squared:  0.4514 
F-statistic: 337.5 on 1 and 408 DF,  p-value: < 2.2e-16

plot(lm.price.log)

The model is still heteroskedastic.

# Visualize the relationship between year and price
ggplot(data=mydf.comp, aes(x=year, y=price)) + geom_point() + theme_classic() + ggtitle("Scatter plot of price vs year") + xlab("Year") + ylab("Price") + theme(axis.text=element_text(size=7))

This scatter graph shows a positive relationship - the newer the car the more expensive it is.

# Numerically justify the graph
cor.test(mydf.comp$year,mydf.comp$price)

    Pearson's product-moment correlation

data:  mydf.comp$year and mydf.comp$price
t = 11.894, df = 408, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.4317940 0.5759648
sample estimates:
      cor 
0.5074219 

This correlation value is closer to 1 suggesting the more recent the year, the more expensive. Using the following hypotheses \(H_0:Price & Mileage are not correlated\) & \(H_1:Price & Mileage are corelated\) where \(\alpha=0.05\) the null hypothesis is very small so I can rject the null hypothesis. Price and year are dependent.

# Create a linear regression model of price and year
lm.price.year <- lm(formula = mydf.comp$price~mydf.comp$year)
summary(lm.price.year)

Call:
lm(formula = mydf.comp$price ~ mydf.comp$year)

Residuals:
   Min     1Q Median     3Q    Max 
-19625  -8413   -545   7247  56495 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)    -2.278e+06  1.939e+05  -11.74   <2e-16 ***
mydf.comp$year  1.144e+03  9.614e+01   11.89   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 10310 on 408 degrees of freedom
Multiple R-squared:  0.2575,    Adjusted R-squared:  0.2557 
F-statistic: 141.5 on 1 and 408 DF,  p-value: < 2.2e-16

plot(lm.price.year)

The r squared value is 26% - not statistically significant, however the intercepts suggest they are very significant because for every unit increase in year, the price reduces by 2.252e-04. The low r-squared could suggest lm is not a good model. Moreover, the Residual vs Fitted plot does not look random and has funnel shape which suggests there is heteroskedasticity.

# Run the model again
lm.year.log <- lm(formula = mydf.comp$price.log~mydf.comp$year)
summary(lm.year.log)

Call:
lm(formula = mydf.comp$price.log ~ mydf.comp$year)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.27462 -0.26062  0.05506  0.28794  2.55329 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)    -88.81052    7.88727  -11.26   <2e-16 ***
mydf.comp$year   0.04906    0.00391   12.55   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4195 on 408 degrees of freedom
Multiple R-squared:  0.2785,    Adjusted R-squared:  0.2767 
F-statistic: 157.5 on 1 and 408 DF,  p-value: < 2.2e-16

plot(lm.year.log)

The model is still heteroskedastic.

# Visualize the relationship between the transmission and price
ggplot(mydf.comp, aes(x=automatic_transmission, y=price)) + geom_boxplot() + theme_minimal() + ggtitle("Transmission type vs Price") + ylab("Price") + xlab("Transmission type (0=Not automatic, 1= Automatic)")

Cars with automatic transmission generally sell at a higher price for a median of 30,000.

# Visualize the relationship between damaged and price
ggplot(mydf.comp, aes(x=damaged, y=price)) + geom_boxplot() + theme_bw() + ggtitle("Price vs damaged") + xlab("Damaged (0=Not damaged, 1= Damaged") + ylab("Price")

Damaged cars are sold around 10,000 cheaper

# Visualize the relationship between the seller type and price
ggplot(mydf.comp, aes(x=first_owner, y=price)) + geom_boxplot() + theme_linedraw() + ggtitle("First owner vs Price") + ylab("Price") + xlab("First owner (0=No, 1=Yes)")

This boxplot shows that cars with only one owner sell at a higher price than those with multiple owners.

# Visualize the relationship between the drivetrain and price
ggplot(mydf.comp, aes(x=drivetrain, y=price)) + geom_boxplot() + theme_minimal() + ggtitle("Drivetrain type vs Price") + ylab("Price") + xlab("Drivetrain type")

Four wheel drive is more expensive followed (with a lot of overlap) by rear wheel drive – although it has a lower mean

# Visualize the relationship between the fuel type and price
ggplot(mydf.comp, aes(x=fuel, y=price)) + geom_boxplot() + theme_minimal() + ggtitle("Fuel type vs Price") + ylab("Price") + xlab("Fuel type")

Electric cars seem to be the most expensive cars while GPL cars are the cheapest. Most of the overlap is between cheaper hybrids and expensive petrol cars.

# Visualize the relationship between max_mpg and price
ggplot(data=mydf.comp, aes(x=max_mpg, y=price)) + geom_point() + theme_classic() + ggtitle("Scatter plot of price vs maximum miles per gallon (mpg)") + xlab("Maximum mpg") + ylab("Price") + geom_smooth(method = "lm", se = FALSE)

Though there is no particular trend, it could be argued there is a downward trend where price decreases as maximum mpg increases.

# Visualize the relationship between min_mpg and price
ggplot(data=mydf.comp, aes(x=min_mpg, y=price)) + geom_point() + theme_classic() + ggtitle("Scatter plot of price vs minimum miles per gallon (mpg)") + xlab("Minimum mpg") + ylab("Price") + geom_smooth(method = "lm", se = FALSE)

The same argument can be made that the larger the minimum mpg, the less expensive the car.

# Visualize the relationship between heated seats, price and third row seating
ggplot(mydf.comp, aes(x=heated_seats, y=price, fill=third_row_seating)) + geom_boxplot() + theme_bw() + ggtitle("Price vs heated seats and third row seating") + xlab("Heated seats") + ylab("Price")

Third row seating increases the price, there’s little difference between heated third row seating but a large difference between no third row or heated seats and unheated third seats.

# Visualize the relationship between bluetooth, price and navigation
ggplot(mydf.comp, aes(x=bluetooth, y=price, fill=navigation_system)) + geom_boxplot() + theme_bw() + ggtitle("Price vs bluetooth and navigation sysetm") + xlab("Bluetooth") + ylab("Price")

Bluetooth increases cost but the combination of no bluetooth and a navigation system is more expensive that both bluetooth and navigation.

# Visualize the relationship between engine_size and price
ggplot(data=mydf.comp, aes(x=engine_size, y=price)) + geom_point() + theme_classic() + ggtitle("Scatter plot of price vs engine_size") + xlab("Engine size") + ylab("Price")

Larger engine sizes are generally more expensive, but there are a lot of expensive cars with smaller engines. 2 litre engines seem the most popular.

# Visualize the relationship between brand and price

ggplot(data=mydf.comp, aes(x=brand, y=price)) + geom_point() + theme_classic() + ggtitle("Scatter plot of price vs brand") + xlab("brand") + ylab("Price") + theme(axis.text=element_text(size=3.7))

Although most brands operate within the same price, some brand cater mid-to-high prices(Alfa, Porsche, Maserati), while others are more budget friendly (FIAT, Mitsubushi, Mazda)

# Visualize 'brand' frequencies
table(mydf.comp$brand)      # To find frequencies

         Alfa          Audi           BMW      Cadillac     Chevrolet          FIAT          Ford         Honda 
           24            21            17            19            12            24            13            24 
      Hyundai        Jaguar          Jeep           Kia          Land         Lexus      Maserati         Mazda 
           18            14            11            20             9            15            12            21 
Mercedes-Benz          MINI    Mitsubishi        Nissan       Porsche        Suzuki        Toyota    Volkswagen 
           15            21            24            13             9             2            10            21 
        Volvo 
           21 

brand.count <- c(24,21,17,19,12,24,13,24,18,14,11,20,9,15,12,21,15,21,24,13,9,2,10,21,21)
brand.name <- c("Alfa", "Audi", "BMW", "Cadillac", "Chevrolet", "FIAT", "Ford", "Honda", "Hyundai", "Jaguar", "Jeep", "Kia", "Land","Lexus","Maserati","Mazda","Mercedes-Benz","MINI","Mitsubishi","Nissan","Porsche","Suzuki","Toyota","Volkswagen","Volvo")

barplot(brand.count, main="Bar chart of brand frequency in 'mydf.comp'", ylab="Frequency in 'mydf.comp'", names.arg=brand.name,  cex.names = 0.7, col="steelblue", density=c(100, 50, 50, 50, 50, 100, 50, 100, 50, 50, 50, 50, 50,50,50,50,50,50,100,50,50,50,50,50,50))

2.3 EDA summary of results

The first scatter plot shows a negative relationship between price and mileage - the more mileage a car has, the cheaper it is. The correlation value was closer to -1 which means the higher the X-axis value (Mileage), the lower the Y-axis value (Price).

If \(H_0:Price & Mileage are not correlated\) & \(H_1:Price & Mileage are corelated\) where \(\alpha=0.05\).

The p-value is: 2.2e-16 which is a lot smaller than the alpha which means I have enough evidence to reject the null hypothesis. Price and mileage are correlated.

The R^2 > 0.5 suggests it is not significant. Still, the p-value suggests they are dependent and the intercept suggests that for every unit change in mileage, the price decreases by 2.008e+00. Visually, the Residual vs. fitted looks like a funnel suggesting heteroskedasticity. The scatterplot suggests a positive relationship - the newer the car the more expensive it is.

This correlation value is closer to 1 suggesting the more recent, the more expensive. Where hypotheses \(H_0:Price & Mileage are not correlated\) & \(H_1:Price & Mileage are corelated\) where \(\alpha=0.05\) the null hypothesis is very small so I can reject the null hypothesis. Price and year are dependent.

This correlation value is closer to 1 suggesting the more recent the year, the more expensive. Using the following hypotheses \(H_0: not correlated\) & \(H_1:Price & Mileage are corelated\) where \(\alpha=0.05\) the null hypothesis is very small so I can reject the null hypothesis. Price and year are dependent.

The model is still heteroskedastic.

2.4 Additional insights and issues

Cars with automatic transmissions generally sell at a higher price for a median of 30,000. Damaged cars are sold around 10,000 cheaper. This boxplot shows that cars with only one owner sell at a higher price than those with multiple owners. Four wheel drive is more expensive followed (with a lot of overlap) by rear wheel drive – although it has a lower mean

Electric cars seem to be the most expensive cars while GPL cars are the cheapest. Most of the overlap is between cheaper hybrids and expensive petrol cars. Though there is no particular trend, it could be argued there is a downward trend where price decreases as maximum mpg increases.

The same argument can be made that the larger the minimum mpg, the less expensive the car. Third-row seating increases the price. Bluetooth increases cost but the combination of no Bluetooth and a navigation system is more expensive than both Bluetooth and navigation.

Larger engine sizes are generally more expensive, but there are a lot of expensive cars with smaller engines. 2-litre engines seem the most popular.

3. Modelling

3.1 Explain your analysis plan

Year and mileage seem to have linear relationships with price (dependent), however they are heteroscedastic and despite transforming the dependent variable using the natural logarithm, the linear regression was still heteroscedastic. To model price as the numeric dependent along with a combination of numeric, categorical and binary independent variables, I will use the linear model function. This will provide Fisher’s p-value which I can compare to my hypothesis and alpha. If larger than alpha, the p-value will confirm the statistical significance of the model. The output will also give coefficients which indicate if they are significant, denoted by ‘’,’’,’’ and how they react in response to the target variable. My approach will be to use Analysis of Covariance (ANCOVA) because of the mixed independent and numeric dependent variables. I will create a maximal multiple linear regression model using the lm() function. I will then use the step() function to find the minimum adequate model. Then I will use the summary() function on the minimum adequate model to understand the significance of each variable. I will also create a logistic regression model and check which explains the price more effectively.

3.2 Build a model for car price

# Build a maximal multiple linear regression model with price as the dependent
maximal.mlr <- lm(mydf.comp$price~.,data = mydf.comp)
summary(maximal.mlr)

# Use the step() function to get the minimum adequate model
model.1 <- step(maximal.mlr)
summary(model.1)
plot(model.1)

# Check for variance inflation factor
vif(maximal.mlr)

vif(model.1)

# Logistic regression
lm.model <- lm(log(mydf.comp$price)~., data = mydf.comp)
summary(lm.model)
step(lm.model)

3.3 Critique model using relevant diagnostics

Fisher’s p-value is smaller than \(\alpha\) which means that the variables in this model affect price, in other words, the price is affected by these variables. The variables that most affect price are brand, where some brands such as Suzuki, Volvo, Maserati, Mercedes-Benz, and FIAT. The price will increase by 1.528e+03 if the car has a navigation system and will decrease by -3.024e+03 if it has Bluetooth. For every price reduction by -4.335e+05 the year will increase by 1.124e+02, while for every price reduction, the minimum mpg increases by 1.337e+02. Akaike’s information criterion has established that the minimum adequate model is: mydf.comp$price ~ brand + year + mileage + drivetrain + min_mpg + max_mpg + navigation_system + bluetooth + heated_seats + price.log

multicollinearity was found in the logistic regression model, so the minimum adequate model’s plots(Residual vs fitting) have a funnel shape suggesting heteroscedasticity while the Q-Q plot looks normal with a slight ‘s’ shape forming due to outliers

3.4 Suggest and implement improvements to your model

Heteroscedasticity can be resolved by transforming the dependent variable using the natural logarithm. One of the principles of conventional linear regression is that constant variance of residuals is broken by heteroscedasticity. Another transformation could be the box cox transformation which involves stabilising non-constant variance and non-normality.

I propose model.1 as the model to predict pricing because Fisher’s p-value makes it statistically significant. It has also been transformed into a minimal adequate model from a maximal model.

4. Modelling another dependent variable

4.1 Model the likelihood of a car being sold by the first owner (using the first_owner variable provided).

Provide a plan of analysis based on relevant EDA for this attribute

# Visualize first owner and price
ggplot(mydf.comp, aes(x=first_owner, y=price)) + geom_boxplot() + theme_linedraw() + ggtitle("First owner vs Price") + ylab("Price") + xlab("First owner (0=No, 1=Yes)")

# Visualize first owner and year
ggplot(mydf.comp, aes(x=first_owner, y=mileage)) + geom_boxplot() + theme_linedraw() + ggtitle("First owner vs Mileage") + ylab("Mileage") + xlab("First owner (0=No, 1=Yes)")

To account model the probability of a binary being true I must use a generalized linear model. I will use all variables and then use the stepwise function to find the minimum adequate model. In order to measure the probability of a car being sold by its first owner I must transform the target variable so that it P=first owner can surpass 1. R automatically calculates the logit. I will then find the odds ratios to better understand the coefficients and explain the most significant variables and their impact on first_owner=1.

Execute the plan

# Maximal generalized linear model
model.2 <- glm(mydf.comp$first_owner ~ mydf.comp$price + mydf.comp$brand + mydf.comp$year + mydf.comp$mileage + mydf.comp$engine_size + mydf.comp$automatic_transmission + mydf.comp$fuel + mydf.comp$drivetrain + mydf.comp$min_mpg + mydf.comp$max_mpg + mydf.comp$damaged + mydf.comp$navigation_system + mydf.comp$bluetooth + mydf.comp$third_row_seating + mydf.comp$heated_seats, data = mydf.comp, family = "binomial")

summary(model.2)

Address any weaknesses of the model and explore methods to improve it

# Find the minimum adequate model
step(model.2)

# Create the minimal adequate model from step
model.3 <- glm(formula = mydf.comp$first_owner ~ mydf.comp$year + mydf.comp$mileage + 
    mydf.comp$engine_size + mydf.comp$automatic_transmission + 
    mydf.comp$min_mpg + mydf.comp$max_mpg + mydf.comp$bluetooth + 
    mydf.comp$third_row_seating + mydf.comp$heated_seats, family = "binomial", 
    data = mydf.comp)

Justify and propose one model. Describe, explain and critique it.

# Summarize significant coefficients
summary(model.2)   

Call:
glm(formula = mydf.comp$first_owner ~ mydf.comp$price + mydf.comp$brand + 
    mydf.comp$year + mydf.comp$mileage + mydf.comp$engine_size + 
    mydf.comp$automatic_transmission + mydf.comp$fuel + mydf.comp$drivetrain + 
    mydf.comp$min_mpg + mydf.comp$max_mpg + mydf.comp$damaged + 
    mydf.comp$navigation_system + mydf.comp$bluetooth + mydf.comp$third_row_seating + 
    mydf.comp$heated_seats, family = "binomial", data = mydf.comp)

Coefficients:
                                        Estimate Std. Error z value Pr(>|z|)    
(Intercept)                           -5.774e+02  1.537e+02  -3.758 0.000172 ***
mydf.comp$price                       -1.324e-05  2.886e-05  -0.459 0.646406    
mydf.comp$brandAudi                    1.052e+00  7.728e-01   1.362 0.173236    
mydf.comp$brandBMW                     4.872e-01  8.353e-01   0.583 0.559751    
mydf.comp$brandCadillac                2.685e-01  8.091e-01   0.332 0.739963    
mydf.comp$brandChevrolet               1.251e+00  1.212e+00   1.032 0.301907    
mydf.comp$brandFIAT                    9.916e-01  7.952e-01   1.247 0.212406    
mydf.comp$brandFord                    4.050e+00  1.330e+00   3.046 0.002322 ** 
mydf.comp$brandHonda                   1.157e+00  8.214e-01   1.409 0.158913    
mydf.comp$brandHyundai                 2.924e-01  8.212e-01   0.356 0.721824    
mydf.comp$brandJaguar                  2.238e+00  1.001e+00   2.237 0.025316 *  
mydf.comp$brandJeep                    1.830e+00  1.078e+00   1.698 0.089506 .  
mydf.comp$brandKia                     9.998e-02  7.819e-01   0.128 0.898254    
mydf.comp$brandLand                    7.259e-01  1.121e+00   0.648 0.517134    
mydf.comp$brandLexus                   6.064e-01  8.645e-01   0.701 0.483019    
mydf.comp$brandMaserati               -1.062e+00  1.231e+00  -0.863 0.388045    
mydf.comp$brandMazda                   1.425e+00  8.519e-01   1.673 0.094408 .  
mydf.comp$brandMercedes-Benz           1.282e+00  9.381e-01   1.367 0.171592    
mydf.comp$brandMINI                    9.684e-01  8.397e-01   1.153 0.248810    
mydf.comp$brandMitsubishi              6.715e-01  8.062e-01   0.833 0.404886    
mydf.comp$brandNissan                  4.970e-01  1.003e+00   0.496 0.620084    
mydf.comp$brandPorsche                 6.171e-01  1.065e+00   0.579 0.562480    
mydf.comp$brandSuzuki                 -1.280e+01  1.664e+03  -0.008 0.993861    
mydf.comp$brandToyota                  9.743e-01  1.127e+00   0.865 0.387281    
mydf.comp$brandVolkswagen             -3.929e-01  7.833e-01  -0.502 0.615936    
mydf.comp$brandVolvo                   8.588e-01  7.943e-01   1.081 0.279580    
mydf.comp$year                         2.884e-01  7.650e-02   3.769 0.000164 ***
mydf.comp$mileage                     -2.110e-05  6.904e-06  -3.056 0.002245 ** 
mydf.comp$engine_size                 -3.493e-01  2.256e-01  -1.548 0.121553    
mydf.comp$automatic_transmission1      1.383e+00  6.703e-01   2.063 0.039099 *  
mydf.comp$fuelElectric                -7.936e-01  1.604e+00  -0.495 0.620677    
mydf.comp$fuelGPL                     -1.847e+01  1.194e+03  -0.015 0.987656    
mydf.comp$fuelHybrid                  -9.734e-01  1.230e+00  -0.791 0.428723    
mydf.comp$fuelPetrol                  -1.303e+00  1.086e+00  -1.200 0.230197    
mydf.comp$drivetrainFront-wheel Drive  7.243e-02  4.055e-01   0.179 0.858213    
mydf.comp$drivetrainRear-wheel Drive  -4.419e-01  4.727e-01  -0.935 0.349878    
mydf.comp$min_mpg                      2.132e-01  8.007e-02   2.663 0.007738 ** 
mydf.comp$max_mpg                     -2.053e-01  7.468e-02  -2.750 0.005969 ** 
mydf.comp$damaged1                    -1.072e-01  3.226e-01  -0.332 0.739544    
mydf.comp$navigation_system1           7.963e-02  3.537e-01   0.225 0.821887    
mydf.comp$bluetooth1                  -1.489e+00  6.670e-01  -2.232 0.025591 *  
mydf.comp$third_row_seating1           7.729e-01  4.425e-01   1.746 0.080743 .  
mydf.comp$heated_seats1               -7.676e-01  3.253e-01  -2.360 0.018291 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 568.22  on 409  degrees of freedom
Residual deviance: 358.54  on 367  degrees of freedom
AIC: 444.54

Number of Fisher Scoring iterations: 15

# Summarize significant coefficients
summary(model.3)    

Call:
glm(formula = mydf.comp$first_owner ~ mydf.comp$year + mydf.comp$mileage + 
    mydf.comp$engine_size + mydf.comp$automatic_transmission + 
    mydf.comp$min_mpg + mydf.comp$max_mpg + mydf.comp$bluetooth + 
    mydf.comp$third_row_seating + mydf.comp$heated_seats, family = "binomial", 
    data = mydf.comp)

Coefficients:
                                    Estimate Std. Error z value Pr(>|z|)    
(Intercept)                       -5.129e+02  1.158e+02  -4.428 9.51e-06 ***
mydf.comp$year                     2.558e-01  5.742e-02   4.455 8.40e-06 ***
mydf.comp$mileage                 -1.814e-05  5.366e-06  -3.380 0.000724 ***
mydf.comp$engine_size             -3.375e-01  1.532e-01  -2.203 0.027622 *  
mydf.comp$automatic_transmission1  8.562e-01  5.862e-01   1.461 0.144131    
mydf.comp$min_mpg                  2.168e-01  6.623e-02   3.273 0.001064 ** 
mydf.comp$max_mpg                 -1.990e-01  6.125e-02  -3.250 0.001155 ** 
mydf.comp$bluetooth1              -1.105e+00  5.811e-01  -1.901 0.057283 .  
mydf.comp$third_row_seating1       8.771e-01  3.829e-01   2.290 0.021997 *  
mydf.comp$heated_seats1           -8.547e-01  2.746e-01  -3.113 0.001854 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 568.22  on 409  degrees of freedom
Residual deviance: 392.45  on 400  degrees of freedom
AIC: 412.45

Number of Fisher Scoring iterations: 5

# Find odds ratio to interpret coefficients
exp(coef(model.3))

                      (Intercept)                    mydf.comp$year                 mydf.comp$mileage 
                    1.857854e-223                      1.291497e+00                      9.999819e-01 
            mydf.comp$engine_size mydf.comp$automatic_transmission1                 mydf.comp$min_mpg 
                     7.135350e-01                      2.354160e+00                      1.242067e+00 
                mydf.comp$max_mpg              mydf.comp$bluetooth1      mydf.comp$third_row_seating1 
                     8.195105e-01                      3.312720e-01                      2.403891e+00 
          mydf.comp$heated_seats1 
                     4.254319e-01 

For every one unit increase in the coefficient, the odds of being the first owner decrease (if the value is below 1) or increase (if it is above 2). For example, For every one unit increase in the brand being volkwagen, the odds of selling the car as a first owner increases by 7.166632e-01.

The likelihood of being a first_owner (versus not being the first_owner) increase by a factor of 2.382e-0.1 for every year. The log odd’s of first_owner must be linear with the other independent variables to stabalise non constant variation and non-normal data. The model has undergone a maximal model and then used Akaike’s information criterion to create the minimal adequate model, which show how important year, mileage and heated seats are to selling as a first owner.